Find \(\tau_d\)-rigid pairs for Nakayama algebras

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This website intends to help illustrate the paper \(\tau_d\)-tilting theory for Nakayama algebras [RV24].

Preliminary

We are working with linear Nakayama algebras with homogeneous relations, admitting a \(d\)-cluster tilting subcategory. Specifically, for an algebraically closed field \(\mathbf{k}\), we are working with the algebras \(\Lambda(n,l)=\mathbf{k}\mathbb{A}_n/R^l\), where \(R\) is the arrow ideal of the quiver \(\mathbb{A}_n\) given by $$ n\longrightarrow n-1 \longrightarrow \cdots \longrightarrow 2 \longrightarrow 1 .$$ The indecomposable modules of \(\operatorname{mod}\Lambda\) are uniquely given through their support on \(\mathbb{A}_n\), which can be described as subintervals of \([1,n]\). Hence, they will be denoted by \(M(a,b)\), where \([a,b]\) is their support-interval. Having this description, we can easily describe the morphism spaces between two indecomposables as follows:

Let \(M(a,b)\) and \(M(c,d)\) be indecomposable \(\Lambda(n,l)\)-modules. The following conditions are equivalent:

1. \(\operatorname{Hom}_\Lambda(M(a,b),M(c,d))\neq 0\)
2. \(\operatorname{Hom}_\Lambda(M(a,b),M(c,d))= \mathbf{k}\)
3. \(b\in [c,d]\) and \(c\in [a,b]\)

We will be making use of the following classification given by Laertis Vaso in [Vas19]:

Let \(\Lambda=\Lambda(n,l)\). There exists a \(d\)-cluster tilting subcategory \(\mathcal{C}\subseteq \operatorname{mod}\Lambda\) if and only if there exists \(p> 1\) such that $$ n=(p-1)\left(\frac{d-1}{2}l+1\right)+\frac{l}{2} $$ and either

1. \(l=2\), or
2. \(l>2\) and both \(d\) and \(p\) are even

holds. The \(d\)-cluster tilting subcategory is then given by $$ \mathcal{C}=\mathrm{add}\left(\bigoplus_{r=0}^{p-1}\tau_d^{-r}(\Lambda)\right). $$

The purpose of this page is to get familiar with \(\tau_d\)-rigid pairs of \(\Lambda(n,l)\), so let us also state the definition:

A pair \((M,P)\in \mathcal{C}\times \operatorname{proj}\Lambda\) is a \(\tau_d\)-rigid pair provided that

1. \(\operatorname{Hom}_\Lambda(M,\tau_d M)=0\), and
2. \(\operatorname{Hom}_\Lambda(P,M)=0\).

In [RV24, Theorem A] we show that the maximal amount of summands a \(\tau_d\)-rigid pair of \(\Lambda(n,l)\) can have is \(n\). We invite to try find different pairs attaining this limit in the graph below.

Functionality of the graph

Below you can generate an algebra \(\Lambda(n,l)\) which admit a \(d\)-cluster tilting subcategory, by choosing \(d\), \(p\) and \(l\) appropriately. The module category will be represented as it's AR-quiver where each vertex is an indecomposable. The darker vertices are indecomposables in the \(d\)-cluster tilting subcategory.

In the graph of the AR-quiver you can construct your own \(\tau_d\)-rigid pair, summand by summand. You add summands by either

1. pressing on a node and selecting if it should be added as a summand of \(M\) or \(P\) in the menu that appears,
2. selecting a node and add it in the menu below, or
3. list the summands in \(M\) and summands in \(P\).

Nodes selected as a summand of \(M\) will be colored red, and nodes selected as a summand of \(P\) will be colored blue. Note that only projective nodes can be added to \(P\) and that the previously selected nodes puts restrictions on which nodes you can choose next. The information on what can't be chosen will be displayed as follows.

• Nodes labeled ⊗ can't be chosen as summand of \(P\),
• nodes labeled × can't be chosen as a summand of \(M\), and
• nodes labeled ⊠ can't be chosen as a summand of \(M\) nor of \(P\).